arXiv subjects
Publications and source records attributed to Emmanuel Grenier.
In this paper, we study the large time behavior of solutions to the linearized Hartree equation near a stable equilibrium which is homogeneous in space and investigate the quantum Landau damping. This allows us to get time decay of the electric field, uniformly in the Planck constant, as well as the uniform-in-time convergence of the quantum density towards the classical density as the Planck constant converges to 0.
We study the Couette Taylor instabilities for an incompressible viscous fluid between two coaxial cylinders of nearly equal radii, allowing counter-rotation with the ratio of rotation rate $\mu \in [-1,1]$. Working in a rotating frame and in a small-gap and small-viscosity regime, we derive the corresponding limiting Navier Stokes system and analyze the linear stability of the Couette flow. In particular, we numerically compute the critical Taylor number for general perturbations and identify a transition near $\mu_c \approx -0.8$: for $\mu > \mu_c$ the most unstable mode is axisymmetric, whereas for $\mu < \mu_c$ the most unstable mode is non-axisymmetric. Near criticality, slowly varying traveling waves are governed by a time-independent Ginzburg Landau equation. The nonlinear coefficient changes sign near $\hat{\mu}_c \approx -0.65$, yielding a supercritical regime for $\mu > \hat{\mu}_c$ and a subcritical regime for $\mu_c < \mu < \hat{\mu}_c$. In the subcritical range, we classify small-amplitude steady states, including Taylor vortex flows, wavy vortices, a two-parameter family of quasi-periodic flows, and a localized traveling perturbation of the Couette flow.
It is well-known that shear flows in a strip or in the half plane are unstable for the incompressible Navier-Stokes equations if the viscosity $\nu$ is small enough, provided the horizontal wave number $\alpha$ lies in a small interval, between the so called lower and upper marginal stability curves. Moreover, a Hopf bifurcation occurs at the upper marginal stability curve. In this article, for various shear flows, we give numerical evidences that this bifurcation is subcritical.
The Couette-Taylor instability occurs in a viscous fluid confined between two coaxial rotating cylinders. When the Taylor number surpasses a critical value, the stable Couette flow destabilizes, giving way to steady Taylor vortices. As the Taylor number increases further, these vortices themselves become unstable, transitioning into wavy Taylor vortices. In this article, we focus on the small-gap limit, where the ratio of the cylinder radii approaches unity and the rotation rates of the cylinders are nearly identical. We provide a rigorous proof of the existence of a critical Taylor number $T_c$, at which the Couette flow loses stability. For Taylor numbers just above $T_c$, under fixed axial periodicity, the solutions to the limiting Navier-Stokes system are governed by a Ginzburg-Landau-type partial differential equation. Beyond the classical Taylor vortex flow, we demonstrate that a two-parameter family of solutions emerges at criticality for $T>T_c$. This family includes not only wavy vortices but also a variety of other exotic flow patterns, all of which remain steady in the frame rotating at the average angular velocity of the cylinders.
The aim of this article is to prove that the linear stability or instability of small Bernstein-Green-Kruskal (BGK) waves is determined by the sign of the derivative of their energy distributions at $0$ energy.
It is well-known that shear flows in a strip or in the half plane are unstable for the Navier-Stokes equations if the viscosity $\nu$ is small enough, provided the horizontal wave number $\alpha$ lies in a small interval, between the so called lower and upper marginal stability curves. The corresponding instabilities are called Tollmien-Schlichting waves. In this letter, we give a simple presentation of the dispersion relation of these waves and study its mathematical properties.
It is well-established that shear flows are linearly unstable provided the viscosity is small enough, when the horizontal Fourier wave number lies in some interval, between the so-called lower and upper marginally stable curves. In this article, we prove that, under a natural spectral assumption, shear flows undergo a Hopf bifurcation near their upper marginally stable curve. In particular, close to this curve, there exists space periodic traveling waves solutions of the full incompressible Navier-Stokes equations. For the linearized operator, the occurrence of an essential spectrum containing the entire negative real axis causes certain difficulties which are overcome. Moreover, if this Hopf bifurcation is super-critical, these time and space periodic solutions are linearly and nonlinearly asymptotically stable.
In this article, we prove that the threshold of instability of the classical Couette flow in $H^s$ for large $s$ is $ν^{1/2}$. The instability is completely driven by the boundary. The dynamic of the flow creates a Prandtl type boundary layer of width $ν^{1/2}$ which is itself linearly unstable. This leads to a secondary instability which in turn creates a sub-layer.
The Rayleigh equation, which is the linearized Euler equations near a shear flow in vorticity formulation, is a key ingredient in the study of the long time behavior of solutions of linearized Euler equations, in the study of the linear stability of shear flows for Navier-Stokes equations and in particular in the construction of the so called Tollmien-Schlichting waves. It is also a key ingredient in the study of vorticity depletion. In this article we locally describe the solutions of Rayleigh equation near critical points of any order of degeneracy, and link their values on the boundary with their behaviors at infinity.
In this article, thanks to a new and detailed study of the Green's function of Rayleigh equation near the extrema of the velocity of a shear layer, we obtain optimal bounds on the asymptotic behaviour of solutions to the linearized incompressible Euler equations both in the whole plane, the half plane and the periodic case, and improve the description of the so called "vorticity depletion property" discovered by F. Bouchet and H. Morita by putting into light a localization property of the solutions of Rayleigh equation near an extremal velocity.
In 1904, Prandtl introduced his famous boundary layer in order to describe the behavior of solutions of Navier Stokes equations near a boundary as the viscosity goes to $0$. His Ansatz has later been justified for analytic data by R.E. Caflisch and M. Sammartino. In this paper, we prove that his expansion is false, up to $O(ν^{1/4})$ order terms in $L^\infty$ norm, in the case of solutions with Sobolev regularity, even in cases where the Prandlt's equation is well posed in Sobolev spaces. In addition, we also prove that monotonic boundary layer profiles, which are stable when $ν= 0$, are nonlinearly unstable when $ν> 0$, provided $ν$ is small enough, up to $O(ν^{1/4})$ terms in $L^\infty$ norm.
This paper is devoted to the study of the nonlinear instability of shear layers and of Prandtl's boundary layers, for the incompressible Navier Stokes equations. We prove that generic shear layers are nonlinearly unstable provided the Reynolds number is large enough, or equivalently provided the viscosity is small enough. We also prove that, generically, Prandtl's boundary layer analysis fails for initial data with Sobolev regularity. In both cases we give an accurate description of the first instability which arises. In some cases a secondary instability appears, leading to several sublayers and to an unexpected complexity of the flow.
The aim of this paper is to describe the long time behavior of solutions of linearized Navier Stokes equations near a concave shear layer profile in the long waves regime, namely for small horizontal Fourier variable $α$, when the viscosity $ν$ vanishes. We show that the solutions converge exponentially to $0$, except in some range of $α$, namely for $ν^{1/4} \lesssim |α| \lesssim ν^{1/6}$, where there exists one unique unstable mode, with an associated eigenvalue $λ$, such that $\Re λ$ is of order $ν^{1/4}$. In this regime we give a complete description of the solutions of linearized Navier Stokes equations as the sum of the projection over the unique exponentially growing mode and of an exponentially decaying term. The study of this linear instability is a key point in the study of the nonlinear instability of Prandtl bounday layers and of shear layer profiles.
This article gathers notes of two lectures given at Grenoble's University in June $2023$, and is an introduction to recent works on shear layers, in collaboration with D. Bian, Y. Guo, T. Nguyen and B. Pausader.
In this paper we study the nonlinear stability of a shear layer profile for Navier Stokes equations near a boundary. This question plays a major role in the study of the inviscid limit of Navier Stokes equations in a bounded domain as the viscosity goes to $0$. The stability of a shear layer for Navier Stokes equations depends on its stability for Euler equations. If it is linearly unstable for Euler, then it is known that it is also nonlinearly unstable for Navier Stokes equations provided the viscosity is small enough: an initial perturbation grows until it reaches $O(1)$ in $L^\infty$ norm. If it is linearly stable for Euler, the situation is more complex, since the viscous instability is much slower, with growth rates of order $O(ν^{-1/4})$ only (instead of $O(1)$ in the first case). It is not clear whether linear instabilities fully develop till they reach a magnitude of order $O(1)$ or whether they are damped by the nonlinearity and saturate at a much smaller magnitude, or order $O(ν^{1/4})$ for instance. In this paper we study the effect of cubic interactions on the growth of the linear instability. In the case of the exponential profile and Blasius profile we obtain that the nonlinearity tame the linear instability. We thus conjecture that small perturbations grow until they reach a magnitude $O(ν^{1/4})$ only, forming small rolls in the critical layer near the boundary. The mathematical proof of this conjecture is open.
We investigate a simple velocity jump process in the regime of large deviation asymptotics. New velocities are taken randomly at a constant, large, rate from a Gaussian distribution with vanishing variance. The Kolmogorov forward equation associated with this process is the linear BGK kinetic transport equation. We derive a new type of Hamilton-Jacobi equation which is nonlocal with respect to the velocity variable. We introduce a suitable notion of viscosity solution, and we prove well-posedness in the viscosity sense. We also prove convergence of the logarithmic transformation towards this limit problem. Furthermore, we identify the variational formulation of the solution by means of an action functional supported on piecewise linear curves. As an application of this theory, we compute the exact rate of acceleration in a kinetic version of the celebrated Fisher-KPP equation in the one-dimensional case.
In this paper, we give an elementary proof of the nonlinear Landau damping for the Vlasov-Poisson system near Penrose stable equilibria on the torus $\mathbb{T}^d \times \mathbb{R}^d$ that was first obtained by Mouhot and Villani in \cite{MV} for analytic data and subsequently extended by Bedrossian, Masmoudi, and Mouhot \cite{BMM} for Gevrey-$γ$ data, $γ\in(\frac13,1]$. Our proof relies on simple pointwise resolvent estimates and a standard nonlinear bootstrap analysis, using an ad-hoc family of analytic and Gevrey-$γ$ norms.
The aim of this paper is to give a detailed presentation of long wave instabilities of shear layers for Navier Stokes equations, and in particular to give a simple and easy to read presentation of the study of Orr Sommerfeld equation and to detail the analysis of its adjoint. Using these analyses we prove the existence of long wave instabilities in the case of slowly rotating fluids, slightly compressible fluids and for Navier boundary conditions, under smallness conditions.